Sojourn-Time Distribution for Geo/Ga,b/1 Queue with Batch Service

This article provides a simple approach for finding the sojourn time of a random customer in a G e o / G a , b / 1 queue with batch service of minimum size a and maximum size b . The service time of each batch assumes a general distribution independent of the arrival process. We find an alternate way to compute the queue-length distribution at post-departure and random epochs. After establishing the functional relationship between the probability-generating function for the queue-length distribution and sojourn-time distribution, they are inverted using the roots of the underlying characteristic equation. We give two closed-form expressions for sojourn-time distributions by applying the underlying characteristic equation’s inside and outside roots. Also, we determine the probability mass function of sojourn time for any specific slot. We deal with a few special cases in our model. Numerical examples for various service-time distributions are performed to exemplify the analysis.